Abstract Algebra II Assignments
Note: Problems labeled PHD are
required for the Ph.D. students and optional for Master’s students.
Assignments now ordered with newest at top!
Due End of Semester
I. Read Section 56
II. Do Section 56 #1, 5 (hint, see problem 4)
III. Show that every polynomial of degree less than 5 over a
field F of characteristic zero is solvable by radicals. Hint: you may use a)
the "if" part of the theorem stated on page 471 in number 1 (which
isn't proven in the book) and b) the fact that
the smallest nonabelian simple group is A5.
II. Do enough of 15.39 to convince yourself that An is simple if n>4.
Due – 12/1/16
I. Read Section 54
II. Do Section 54 #4, 5, 6, 9a
III. Read Section 35
IV. Do Section 35 #4, 18, 19, 26, 29 (you may use the facts stated in class and in problems 27 and 28 without proving them)
Due – 11/17/16
I. Read Section 53
II. Do 53 #1-8, 11, 22, 23
Due – 11/3/16
I. Read Section 51
II. Do 51#1, 3, 4, 9, 11, 13
Due 10/27/2016
I. Read Section 50
II. Do 50 #2, 4, 6, 7, 8, 9, 10, 22, 23, 21 (I think these will be easier in this order)
Due 10/20/16
I. Read Section 49
II. Do 49#1, 2, 4, 5, 7, 9, 11
Due 10/13
I. Do 48 # 4, 5, 7, 10, 11, 15, 17, 18, 20, 32, 39 (hint for 39: show that any automorphism of the real numbers must fix the rational numbers; you might also want to use some properties of the reals you know from analysis)
II. PHD (though recommended for everyone): Do 48 #36, 37
III. Start reading Section 49
Due 10/6
Part A
I. Do Section 34 #1, 3, 5, 8
II. a. A group G is metabelian if it contains a subgroup N such that N and G/N are both abelian. Show that if G is metabelian then any subgroup H of G is also metabelian.
b. Bonus question: If H is a normal subgroup of G and G is metabelian, is G/H metabelian?
III. Start reading Section 48
Due 9/27
I. Do Section 33 #1, 2, 3, 9, 11,
12
II. Read Section 34
Due 9/22
I. Do Section 32 #3, 4, 5, 6, 8
Due 9/20/2016
I. Do Section 31 #32, 33, 37
II. PHD: Do 31 #38
III. Read Section 32
Due 9/15/2016
I. Do 31 #3, 4, 5, 22, 23 (note: "be prepared to justify your answers" means "justify your answers.")
Due 9/13/2016
I. Read Section 31
II. Do Section 30 #4, 5, 9, 10, 24, 26
III. PHD: Do Section 30 #27
Due 9/8/2016
I. Read Section 30
II. Do Section 29 #6, 12, 14, 16,
30, 31, 34
III. PHD: Do Section 29 #35
Due 9/6/2016
I. Do Section 29 #1, 2, 4, 29, 33
II. PHD: Do Section 29 #36
Due 9/1/2016
I. Read section 29
II. Do Section 27 #5, 6, 14, 18, 19, 30, 31
III. PHD: Show that there are ideals of Z[x] that are not principal.
Due 8/30/2016
I. Finish reading Section 27 (pages 249-252)
II. Let N1 and N2 be two prime ideals of the commutative ring R. Show that the intersection of N1 and N2 is not necessarily a prime ideal.
III. Do Section 27 #1, 24 (hint: use a theorem from earlier in the book – I think this is difficult to do directly), 25, 26, 27, 28
Due 8/25/2016
I. Read Section 26
II. Do Section 26 #4, 8, 9, 18, 22, 24, 26
III. Do these:
a. Let R be the real numbers, and let S=R[x]/<2x2-4x+6>. Compute (and simplify) (x2 -1)2 in S by finding a coset representative of degree less than 4.
b. Show that for any commutative ring R, R[x]/<x> is isomorphic to R.
IV. Start Reading Section 27 (pages 245-248)